December  2017, 22(10): 3891-3901. doi: 10.3934/dcdsb.2017200

A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance

Department of Mathematics, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan

Received  April 2014 Revised  July 2017 Published  August 2017

The initial value problem for a reaction-diffusion system with discontinuous nonlinearities proposed by Hofbauer in 1999 as an equilibrium selection model in game theory is studied from the viewpoint of the existence and stability of solutions. An equilibrium selection result using the stability of a constant stationary solution is obtained for finite symmetric 2 person games with a 1/2-dominant equilibrium.

Citation: Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200
References:
[1]

H. Deguchi, Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1139-1167.  doi: 10.1017/S0308210500004315.  Google Scholar

[2]

H. Deguchi, On weak solutions of parabolic initial value problems with discontinuous nonlinearities, Nonlinear Anal., 63 (2005), e1107-e1117.  doi: 10.1016/j.na.2004.10.004.  Google Scholar

[3]

H. Deguchi, Existence, uniqueness and stability of weak solutions of parabolic systems with discontinuous nonlinearities, Monatsh. Math., 156 (2009), 211-231.  doi: 10.1007/s00605-008-0082-y.  Google Scholar

[4]

H. Deguchi, Weak solutions of a parabolic system with a discontinuous nonlinearity, Nonlinear Anal., 71 (2009), e2902-e2911.  doi: 10.1016/j.na.2009.07.005.  Google Scholar

[5]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.  Google Scholar

[6]

J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA, 1988.  Google Scholar

[7]

J. Hofbauer, Stability for the best response dynamics, preprint, 1995. Google Scholar

[8]

J. Hofbauer, Equilibrium selection via travelling waves, in Game Theory, Experience, Rationality (eds. W. Leinfellner and E. Köhler), Kluwer Academic Publishers, Dordrecht, 5 (1998), 245-259.   Google Scholar

[9]

J. Hofbauer, The spatially dominant equilibrium of a game, Ann. Oper. Res., 89 (1999), 233-251.  doi: 10.1023/A:1018979708014.  Google Scholar

[10]

J. Hofbauer and P. L. Simon, An existence theorem for parabolic equations on $\textbf{R}^N$ with discontinuous nonlinearity, Electron. J. Qual. Theory Differ. Equ., 8 (2001), 1-9.   Google Scholar

[11]

H. P. McKean and V. Moll, Stabilization to the standing wave in a simple caricature of the nerve equation, Comm. Pure Appl. Math., 39 (1986), 485-529.  doi: 10.1002/cpa.3160390403.  Google Scholar

[12]

S. MorrisR. Rob and H. S. Shin, p-dominance and belief potential, Econometrica, 63 (1995), 145-157.  doi: 10.2307/2951700.  Google Scholar

[13]

D. Oyama, S. Takahashi and J. Hofbauer, Monotone Methods for Equilibrium Selection Under Perfect Foresight Dynamics, Working paper No. 0318, University of Vienna, 2003. doi: 10.2139/ssrn.779864.  Google Scholar

[14]

S. Takahashi, Perfect foresight dynamics in games with linear incentives and time symmetry, Internat. J. Game Theory, 37 (2008), 15-38.  doi: 10.1007/s00182-007-0101-6.  Google Scholar

[15]

D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129.  doi: 10.1137/0514086.  Google Scholar

show all references

References:
[1]

H. Deguchi, Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1139-1167.  doi: 10.1017/S0308210500004315.  Google Scholar

[2]

H. Deguchi, On weak solutions of parabolic initial value problems with discontinuous nonlinearities, Nonlinear Anal., 63 (2005), e1107-e1117.  doi: 10.1016/j.na.2004.10.004.  Google Scholar

[3]

H. Deguchi, Existence, uniqueness and stability of weak solutions of parabolic systems with discontinuous nonlinearities, Monatsh. Math., 156 (2009), 211-231.  doi: 10.1007/s00605-008-0082-y.  Google Scholar

[4]

H. Deguchi, Weak solutions of a parabolic system with a discontinuous nonlinearity, Nonlinear Anal., 71 (2009), e2902-e2911.  doi: 10.1016/j.na.2009.07.005.  Google Scholar

[5]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.  Google Scholar

[6]

J. C. Harsanyi and R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA, 1988.  Google Scholar

[7]

J. Hofbauer, Stability for the best response dynamics, preprint, 1995. Google Scholar

[8]

J. Hofbauer, Equilibrium selection via travelling waves, in Game Theory, Experience, Rationality (eds. W. Leinfellner and E. Köhler), Kluwer Academic Publishers, Dordrecht, 5 (1998), 245-259.   Google Scholar

[9]

J. Hofbauer, The spatially dominant equilibrium of a game, Ann. Oper. Res., 89 (1999), 233-251.  doi: 10.1023/A:1018979708014.  Google Scholar

[10]

J. Hofbauer and P. L. Simon, An existence theorem for parabolic equations on $\textbf{R}^N$ with discontinuous nonlinearity, Electron. J. Qual. Theory Differ. Equ., 8 (2001), 1-9.   Google Scholar

[11]

H. P. McKean and V. Moll, Stabilization to the standing wave in a simple caricature of the nerve equation, Comm. Pure Appl. Math., 39 (1986), 485-529.  doi: 10.1002/cpa.3160390403.  Google Scholar

[12]

S. MorrisR. Rob and H. S. Shin, p-dominance and belief potential, Econometrica, 63 (1995), 145-157.  doi: 10.2307/2951700.  Google Scholar

[13]

D. Oyama, S. Takahashi and J. Hofbauer, Monotone Methods for Equilibrium Selection Under Perfect Foresight Dynamics, Working paper No. 0318, University of Vienna, 2003. doi: 10.2139/ssrn.779864.  Google Scholar

[14]

S. Takahashi, Perfect foresight dynamics in games with linear incentives and time symmetry, Internat. J. Game Theory, 37 (2008), 15-38.  doi: 10.1007/s00182-007-0101-6.  Google Scholar

[15]

D. Terman, A free boundary problem arising from a bistable reaction-diffusion equation, SIAM J. Math. Anal., 14 (1983), 1107-1129.  doi: 10.1137/0514086.  Google Scholar

Figure 1.  Condition (B2)
Figure 2.  is the set of ${\bf u} \in \Delta$ to which the pure strategy $i$ (i.e., ${\bf e}_i$) is the best reply in the game $(13)$ for $i = 1, 2, 3$. The three dots denote the middle points of the edges.
Figure 3.  is the set of ${\bf u} \in \Delta$ to which the pure strategy $i$ (i.e., ${\bf e}_i$) is the best reply in the game $(14)$ for $i = 1, 2, 3$. The three dots denote the middle points of the edges.
[1]

Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369

[2]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032

[3]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817

[4]

Georg Hetzer. Global existence for a functional reaction-diffusion problem from climate modeling. Conference Publications, 2011, 2011 (Special) : 660-671. doi: 10.3934/proc.2011.2011.660

[5]

Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635

[6]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85

[7]

Patrizia Pucci, Maria Cesarina Salvatori. On an initial value problem modeling evolution and selection in living systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 807-821. doi: 10.3934/dcdss.2014.7.807

[8]

Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4041-4069. doi: 10.3934/dcds.2013.33.4041

[9]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083

[10]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721

[11]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519

[12]

Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805

[13]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[14]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[15]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106

[16]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[17]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[18]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587

[19]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815

[20]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (21)
  • HTML views (5)
  • Cited by (0)

Other articles
by authors

[Back to Top]